What is the meaning behind the terminology "point-set level (object or category)” in context of stable homotopy theory? This appears e.g. in following excerpt quoted from Tom Bachmann's ThesisInvertible Objects in Motivic Homotopy Theory (p 2):
[...] Just as the homotopy category of spaces can be obtained from several “point-set level” categories of spaces, the stable homotopy category SH can be obtained from several categories of “point-set level objects” called spectra by passing to an appropriate equivalence relation on maps, also called weak equivalence.
In the context of homotopy theory the language of $\infty$-categories is used frequently. Building it up in forms of models founded on the theory of ordinary sets and their points is quite elaborate, but it proves to be really worth it. Since many things can be stated quite easily and elegantly in this language, it is common to skip these point-set details and work in a somewhat axiomatic fashion with this language. From this axiomatic point of view, $\infty$-categories exist on their own but can be represented by point-set models. Picking up on Bachmann's example, the stable $\infty$-category of spectra is "just" the stabilization of the $\infty$-category of anima/spaces/homotopy types. But you can model this $\infty$-category as the model category of sequential spectra, or as the model category of symmetric spectra, or as the model category of orthogonal spectra...